The influence of nonlinearities on the propagation of stress waves through soft tissues is currently an open research area. Understanding this influence could yield insights into the damage mechanisms of soft tissues in response to rapid and strong external excitations. In the context of this dissertation, soft tissues are idealized as nonlinear viscoelastic materials, and the focus is on the mechanical aspects of the tissue behavior. Two nonlinear material models are explored. One of them, a nonlinear extension of the standard solid viscoelastic model, is employed first to describe brain tissue behavior, and second, to study the interaction of blast pressure waves with viscoelastic systems. The second material model, obtained through a maximum dissipation, thermodynamically consistent construction, is employed for the studies of longitudinal wave propagation.

In order to focus on the effects of the material nonlinearity, a geometrically fundamental model for longitudinal stress waves is employed. Theoretical studies including dispersion and asymptotic analyses are carried out in order to further the current understanding of wave propagation characteristics, such as the dependence of the wave speed and attenuation on the frequency, the effect of material dissipation, and the nonlinear steepening of wave fronts. Computational studies are carried out to examine various aspects of the nonlinear wave propagation. A unique nonlinear phenomenon related to the steepening of wave fronts is observed: the tissue absorbs energy in a localized fashion at the location of the moving steep wave front. This situation could be potentially detrimental to the tissue. Finally, the interplay between geometry (non--uniform cross--section) and material nonlinearity is studied. It is observed that a contracting cross--section promotes the development of much steeper stress wave fronts. The spatial location at which the steep wave front develops appears to be related to the elapsed time and the amplitude of the external load. Understanding this relationship could help establish a link between the location of the tissue damage and the external loading. This dissertation work can serve as a basis for better understanding the mechanical causes underlying mild traumatic brain injury, for example, as a consequence of head impact or explosive blast waves.